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Limit distributions of the upper order statistics for the Lévy-frailty Marshall-Olkin distribution. (English) Zbl 1469.62252

The Lévy-frailty subfamily of the Marshall-Olkin (LFMO) distribution as a particular case of Marshall-Olkin (MO) distribution turned out to be a flexible and powerful modeling tool that requires few parameters and is easy to simulate. The second section of the article is devoted to some basics. The Marshall-Olkin and Lévy-frailty Marshall-Olkin distributions are defined and a result on marginal distribution of the order statistics of LFMO distribution is shown. The main result of the paper which refers to the asymptotic behavior of the upper order statistics of the LFMO distribution together with generalizations to other multivariate models and results on convergence intervals and convergence rates are presented in the third section of the paper. Some numerical results via Monte Carlo simulations are shown in the fourth section. The proof of the main result and additional computations are contained in the last section of the article as well as in the Appendix.

MSC:

62G30 Order statistics; empirical distribution functions
62G32 Statistics of extreme values; tail inference
62E10 Characterization and structure theory of statistical distributions
60B10 Convergence of probability measures
60K10 Applications of renewal theory (reliability, demand theory, etc.)
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[1] Asmussen, S.: Applied probability and queues. Springer, New York (2003) · Zbl 1029.60001
[2] Asmussen, S., Glynn, P. W.: Stochastic Simulation: Algorithms and Analysis, vol. 57. Springer (2007) · Zbl 1126.65001
[3] Barrera, J.; Ycart, B., Bounds for left and right window cutoffs, ALEA: Latin Amer. J. Probab. Math. Stat., 11, 2, 445-458 (2014) · Zbl 1346.60040
[4] Bernhart, G., Fernández, L., Mai, J. F., Schenk, S., Scherer, M.: A Survey of Dynamic Representations and Generalizations of the Marshall-Olkin Distribution. In: Marshall-Olkin Distributions-Advances in Theory and Applications, pp. 1-13. Springer (2015) · Zbl 1365.62187
[5] Botev, Z.; L’Ecuyer, P.; Simard, R.; Tuffin, B., Static network reliability estimation under the Marshall-Olkin copula, ACM Trans. Model. Comput. Simul., 26, 2, 14 (2016) · Zbl 1369.90057 · doi:10.1145/2775106
[6] Cherubini, U., Durante, F., Mulinacci, S.: Marshall-Olkin Distributions — Advances in Theory and Applications: bologna, vol. 141. Springer, Italy (2015) · Zbl 1322.60002
[7] David, H. A.: Topics in the History of Order Statistics. In: Advances in Distribution Theory, Order Statistics, and Inference, pp. 157-172. Springer (2006) · Zbl 05196669
[8] De Haan, L.; Peng, L., Exact rates of convergence to a stable law, J. Lond. Math. Soc., 59, 3, 1134-1152 (1999) · Zbl 0947.60025 · doi:10.1112/S0024610799007346
[9] Engel, J., Scherer, M., Spiegelberg, L.: One-Factor Lévy-Frailty Copulas with Inhomogeneous Trigger Rates. In: Soft Methods for Data Science, pp. 205-212. Springer (2017)
[10] Fernández, L., Mai, J. F., Scherer, M.: The Mean of Marshall-Olkin-Dependent Exponential Random Variables. In: Marshall-Olkin Distributions-Advances in Theory and Applications, pp. 33-50. Springer (2015) · Zbl 1365.62189
[11] Finkenstadt, B., Rootzén, H. (eds.): Extreme values in finance, telecommunications, and the environment. CRC Press, Boca Raton (2003)
[12] Gudendorf, G., Segers, J.: Extreme-Value Copulas. In: Copula Theory and Its Applications, pp. 127-145. Springer (2010) · Zbl 1349.62207
[13] Gumbel, EJ, Les valeurs extrêmes des distributions statistiques, Ann. l’inst. Henri Poincaré, 5, 2, 115-158 (1935) · Zbl 0011.36102
[14] Hering, C.; Mai, JF, Moment-based estimation of extendible Marshall-Olkin copulas, Metrika, 75, 5, 601-620 (2012) · Zbl 1362.62122 · doi:10.1007/s00184-011-0344-x
[15] Hüsler, J., Multivariate extreme values in stationary random sequences, Stoch. Process. Appl., 35, 1, 99-108 (1990) · Zbl 0703.62027 · doi:10.1016/0304-4149(90)90125-C
[16] Kallenberg, O.: Foundations of modern probability. Springer Science & Business Media (2002) · Zbl 0996.60001
[17] Leadbetter, MR, Extremes and local dependence in stationary sequences, Zeitsch. Wahrscheinlichkeitstheorie Verwandte Gebiete, 65, 2, 291-306 (1983) · Zbl 0506.60030 · doi:10.1007/BF00532484
[18] Leadbetter, M. R., Lindgren, G., Rootzén, H.: Extremes and related properties of random sequences and processes. Springer Science & Business Media (2012) · Zbl 0518.60021
[19] Lindskog, F.; McNeil, AJ, Common Poisson shock models: applications to insurance and credit risk modelling, Astin Bullet., 33, 2, 209-238 (2003) · Zbl 1087.91030 · doi:10.1017/S0515036100013441
[20] Mai, JF; Scherer, M., Lévy-frailty copulas, J. Multivar. Anal., 100, 7, 1567-1585 (2009) · Zbl 1162.62048 · doi:10.1016/j.jmva.2009.01.010
[21] Mai, JF; Scherer, M., Reparameterizing Marshall-Olkin copulas with applications to sampling, J. Stat. Comput. Simul., 81, 1, 59-78 (2011) · Zbl 1206.62101 · doi:10.1080/00949650903185961
[22] Mai, J. F.: Multivariate Exponential Distributions with Latent Factor Structure and Related Topics. Ph.D. thesis, Technische Universität München (2014)
[23] Mai, JF, Extreme-value copulas associated with the expected scaled maximum of independent random variables, J. Multivar. Anal., 166, 50-61 (2018) · Zbl 1398.62130 · doi:10.1016/j.jmva.2018.02.005
[24] Mai, JF; Scherer, M., Sampling exchangeable and hierarchical Marshall-Olkin distributions, Commun. Stat.-Theory Methods, 42, 4, 619-632 (2013) · Zbl 1347.62025 · doi:10.1080/03610926.2011.615437
[25] Mai, J. F., Scherer, M.: Simulating copulas: stochastic models, sampling algorithms, and applications, vol. 6 World Scientific (2017) · Zbl 1367.65002
[26] Maller, R., Schindler, T.: Small time convergence of subordinators with regularly or slowly varying canonical measure, vol. 129. Elsevier (2019) · Zbl 1422.60088
[27] Marichal, JL; Mathonet, P.; Waldhauser, T., On signature-based expressions of system reliability, J. Multivar. Anal., 102, 10, 1410-1416 (2011) · Zbl 1219.62154 · doi:10.1016/j.jmva.2011.05.007
[28] Marshall, A.; Olkin, I., A multivariate exponencial distribution, J. Am. Stat. Assoc., 62, 2014, 30-44 (1967) · Zbl 0147.38106 · doi:10.1080/01621459.1967.10482885
[29] Matus, O., Barrera, J., Moreno, E., Rubino, G.: On the Marshall-Olkin copula model for network reliability under dependent failures, 10.1109/TR.2018.286570710.1109/TR.2018.2865707, vol. 68 (2019)
[30] Nagaraja, H. N.: Order Statistics from Independent Exponential Random Variables and the Sum of the Top Order Statistics. In: Advances in Distribution Theory, Order Statistics, and Inference, pp. 173-185. Springer (2006) · Zbl 05196670
[31] Nolan, J.P.: Stable Distributions - Models for Heavy Tailed Data. Birkhauser, Boston. In progress, Chapter 1 online at http://fs2.american.edu/jpnolan/www/stable/stable.html (2018)
[32] Prokhorov, YV, Convergence of random processes and limit theorems in probability theory, Theory Probab. Appl., 1, 2, 157-214 (1956) · doi:10.1137/1101016
[33] Whitt, W.: Stochastic-Process Limits: an introduction to Stochastic-Process limits and their application to queues. Springer Science & Business Media (2002) · Zbl 0993.60001
[34] Wüthrich, MV, Limit distributions of upper order statistics for families of multivariate distributions, Extremes, 8, 4, 339-344 (2005) · Zbl 1142.60356 · doi:10.1007/s10687-006-0007-x
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